Ambiguous Case Law Of Cosines

We have seen combinations of sides and angles associated with the criteria for congruent triangles.

purpledragon

SAS - Side, Bending, Side
ASA - Angle, Side, Angle
AAS - Bending, Angle, Side
SSS - Side, Side, Side
HL - Hypotenuse Leg for Right Δs

We saw that SSA (Side, Side, Angle) did not work to evidence triangles congruent.
Nosotros politely chosen it the "Donkey Theorem."

These combinations are at present going to show us when to use the Law of Sines and Law of Cosines.
And yes, SSA is however going to be a trouble maker!

Using Law of Cosines with SSA

We accept seen that using the Police of Cosines with the combinations SAS and SSS guarantees 1 unique solution and 1 unique triangle. Working with the third option of SSA, withal, continues to leave the door open for several different situations to occur, as information technology did with the Police of Sines. SSA is withal referred to as the Ambiguous Instance.

Unlike the Ambiguous Case for the Law of Sines, the Ambiguous Case associated with the Law of Cosines will always require the solution to a quadratic equation to find a missing side.
Dig out the old quadratic formula: quadformual

SSA : If two sides and the non-included angle are given, three situations may occur.
When dealing with the Law of Cosines, you will be looking to discover a side.

(1) NO triangle exists - no solution
(2) TWO dissimilar triangles be - two solutions
(3) exactly Ane triangle exists - 1 solution.

SSA - Two sides and the non-included angle are given.

Situation 1: NO triangle exists. In ΔABC, m∠A = 30º, a = 7, and b = xvi. Find c. Solution:	Respond: This triangle does not exist.
The solution shows that this triangle does non exist because there is no real number solution to the quadratic equation. Nosotros need existent number values to represent the sides of a triangle.

SSA - Ii sides and the non-included angle are given.

Situation 2: Ii triangles exist. In ΔABC, m∠A = 30º, a = 10, and b = xvi. Find c. Solution: Since angle A is the merely known angle, cull the Law of Cosines formula that utilizes angle A. Now, use the quadratic formula to solve for c.	Reply: c = 19.8564 or c = 7.8564 There are ii possible triangles.
The quadratic equation solution shows that there are two possible solutions for c, which means at that place are two possible triangles (one acute triangle and one birdbrained triangle).

SSA - Two sides and the non-included bending are given.

Situation iii: Exactly Ane triangle exists. In ΔABC, m∠A = 30º, a = 20, and c = 16. Find c. Solution:	ANSWER: c = 32.1866979 One distinct triangle exists.
There is only 1 possible solution. Since c represents the side of a triangle, it must exist a positive value. The negative solution is rejected.

To summarize the Ambiguous Case:
Dissimilar the Cryptic Case for the Law of Sines with all of its possible situations, the Ambiguous Case for the Law of Cosines leaves the conclusion making on the number of triangles (or solutions) to the quadratic equation.

The solution(s) to the quadratic equation tell you the needed information:
(ane) if the solution is "non Existent", the triangle does not exist (no solution).
(2) if the solution is "two Real positive values", there are two possible triangles (2 solutions).
(3) if the solution is "one positive and one negative Existent values", there is one triangle (one solution).

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